C07 Exponentials and Logarithms
对应大纲的内容
| MM5.1 | $y = a^{x}$ and its graph, for simple positive values of $a$. |
| MM5.2 | Laws of logarithms:
including the special cases:
Questions requiring knowledge of the change of base formula will not be set. |
| MM5.3 | The solution of equations of the form $a^{x}=b$, and equations which can be reduced to this form, including those that need prior algebraic manipulation. For example, $3^{2x}=4$ and $25^{x}-3\times5^{x}+2=0$. |
学习目标
- Master the laws of logarithms by fluently applying product, quotient, and power identities to combine and dismantle complex expressions without relying on the change-of-base formula.
- Execute algebraic base matching to solve exponential equations by manipulating indices to achieve common bases $\left(a^{f(x)}=a^{g(x)}\right)$ or by applying logarithms to isolate unknown exponents.
- Deconstruct disguised exponential quadratics to solve complex non-linear equations (e.g., $k\left(a^x\right)^2+l\left(a^x\right)+m=0$ ) using strategic substitutions $\left(y=a^x\right)$, strictly enforcing the condition $a^x>0$.
- Evaluate logarithmic domain constraints when solving equations to accurately identify and discard extraneous roots that force the argument of any logarithm to be non-positive.
- Analyse inverse geometric architectures by sketching and interpreting the base curves $y=a^x$ and $y=\log _a x$, exploiting their reflectional symmetry across $y=x$ to identify respective horizontal and vertical asymptotes.
建议时间
| Task | No. of Questions | Suggested Time |
|---|---|---|
| Examples | 14 | 120 min |
| Quiz | 7 | 20 min |
| Practice A | 40 | 60 min |
| Practice B | 30 | 75 min |
| Total | 91 | approx. 4-5 hrs |
